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On the Classification of Locally Hamming Distance-Regular Graphs 数理解析研究所講究録 第 76850巻 1991 年 50-61 On the Classification of Locally Hamming Distance-Regular Graphs BY MAKOTO MATSUMOTO Abstract. A distance-regular graph is locally Hamming if it is locally isomorphic to a Hamming scheme $H(r, 2)$ . This paper rediscovers the connection among locally Ham- ming distance-regular graphs, designs, and multiply transitive permutation groups, through which we classify some of locally Hamming distance-transitive graphs. \S 1. Introduction. By a graph we shall mean a finite undirected graph with no loops and no multiple edges. For a graph $G,$ $V(G)$ denotes the vertex set and $E(G)$ denotes the edge set of $G$ . For a vertex $v$ of a graph $G,$ $N(v)$ denotes the set of adjacent vertices with $v$ . By $\ovalbox{\tt\small REJECT}_{2}$ we denote the two-element field, and by $H(r)$ we denote the r-dimensional Hamming scheme for $r\geq 1$ ; that is, $H(r)$ is such a graph that its vertex set is the $\#_{2}^{=r}$ vector space a,nd $\tau\iota,$ $v\in\#_{2}^{=r}$ are adjacent if and only if the Hamming distance $d(u, v)=1$ ; i.e., $\#\{i u_{i}\neq v_{i}\}=1$ where $u=(u_{1}, \ldots u_{r})$ and $v=(v_{1}, \ldots , v_{r})$ . The graph obtained from $H(r)$ by identifying antipodal points is called a folded Hamming sheme(or a folded Hamming cube in [3, p.140]). The graph $H(3)$ is called a cube, and the induced subgraph obtained by removing one vertex tog$e$ ther with the three incident edges from a cube is called a tulip. A tulip contains exactly three vertices with degree two, and these vertices are called petals. The unique vertex which is not adjacent to any petals is called the root of the tulip. A locally Hamming graph $G$ is a connected graph such that (1) $G$ has no triangle, (2) for any $u,$ $v\in V(G)$ satisfying $d(u, v)=2$ , there exist exactly two vertices adjacent to both $u$ and $v$ , $T$ $G$ (3) for any subgraph of which is isomorphic to a tulip with petals $p,$ $q,$ $r$ , $r$ there exists a vertex $x\in V(G)$ adjacent to all $p,$ $q$ ) and . (The uniqueness of $x$ follows from the condition (2).) Of course $H(r)$ is an example of locally Hamming graphs, and it was proved that a distance-regular graph with $parc\urcorner metersa_{1}=0,$ $a_{2}=0,$ $c_{2}=2$ , and $c_{3}=3$ is a locally Hamming $graph[2]$ [$3$ , Lemma 4.3.5]. For a generalization to $H(r,q)$ for $q\geq 3$ , see [10]. (A locally Hamming graph is exactly the same thing with a rectagraph such that any 3-claw determines a unique 3-cube in Brouwer’s terminology[3, p.153].) Our objective is to classify all locally Hamming distance-regular graphs(LHDRG). A number of authors have contributed to this $aim[2][3][5][10]$ . This paper provides an approach to this goal using a universal covering of a LHDRG. Main result is as follows. Typeset by $A_{\mathcal{M}}S- T_{FX}$ 51 MAIN RESULT. Let $G$ be an r-regular $distaJ1ce- l\cdot eg$ ular graph with parameters $a_{1}=$ $a_{2}=0$ and $c_{2}=2,$ $c_{3}=3,$ $ot1_{1}ertl_{la1l}H(r)$ . $Tal_{\overline{1}}e$ th$emi_{1l}imun3nunIbert$ such that either $a_{t}\neq 0$ or $c_{t}\neq t$ occurs. Pu $td=2t+1$ if $c_{t}=t$ , an $d$ put $d=2tott1$erwise. Then, $tAere$ exists a $t-(r, d, /\backslash )$ design witli $/\backslash \leq(r-t)/(d-t)$ . if $G$ is distance-transitive, $s$ $\lfloor(d-1)/2\rfloor$ $d$ $Aut(G)$ contain a -homogeneous $p_{\supset}^{\circ_{loup}}$ of egre$er$ acting on the block set of the $t-(r, d, /\backslash )$ design. COROLLARY. (See Theorem 3). Any distance-transitive graph with $a_{1}=a_{2}=a_{3}=0$ and $c_{i}=i$ for $i=1,2,3,4$ is a Hamming scheme or a folded Hamm$ingsch$eme. For the case $t=2,3$ , some nontrivial examples $are$ listed. \S 2. The universal covering. Let $G,$ $H$ be graphs. A mapping $f$ : $V(H)arrow V(G)$ is said to be a covering if it is surjective and for any $u\in V(H),$ $f_{N(u)}$ is a bijection $N(u)arrow N(f(u))$ . For a vertex $u$ of $G$ , the set $f^{-1}(u)$ is called the fiber on $u$ . In the previous paper[9] we proved the following propositions. PROPOSITION 1. Let $G$ be a locally $Ha$mming graph with valen$cyr$ . Then, there exists a covering $f$ : $H(r)arrow C_{\tau}$ . Let $H$ be a $loc$ally Hamming graph an $d$ le $th$ : $Harrow G$ be a covering. For any vertex $u\in V(G)$ and for any vertices $x\in f^{-1}(u),$ $y\in h^{-1}(u)$ , tliere exists a unique covering $g:H(r)arrow H$ such that $g:xrightarrow y$ and $f=hg$ hold. The above covering $f$ is called a universal covering. We define the fundamental group $\pi(G, f)$ of the pair $G$ and $f$ : $H(r)arrow G$ as $\pi(G, f)=\{\gamma\in Aut(H(r))|f\gamma=f\}$ . This definition depends on the choice of $f$ , but unique upto conjugacy in $Aut(H(r))$ . (Note that this definition is not equivalent to the fundamental group of $G$ as a one- dimensional topological object.) / Let $\Gamma$ be a subgroup of $Aut(H(r))$ . We define the discreteness $d_{\Gamma}$ of I’ by $d_{\Gamma}= \min\{d(u,\gamma u)|\gamma\in\Gamma,\gamma\neq id, u\in V(H)\}$ where $d$ denotes the Hamming distance. For the trivial group {id}, its discreteness $\Gamma$ is defined to be $\infty$ . Let be a subgroup of $Aut(H(r))$ with $d_{\Gamma}\geq 5$ . Then we define the quotient graph $H(r)/\Gamma$ as follows. The vertex set is the set of coset of $V(H(r))$ by $\Gamma$ ; i.e., the set $\{\Gamma u|u\in H(r)\}$ , where $\Gamma u=\{\gamma u|\gamma\in\Gamma\}$ . The two vertices $\Gamma u,$ $\Gamma v$ are adjacent if and only if there exists a vertex $w\in\Gamma u$ such that $w$ is adjacent with $v$ . The obtained graph was proved to be locally Hamming. The canonical mapping $f$ : $H(r)arrow H(r)/\Gamma$ defined by $f$ : $u\mapsto\Gamma u$ is a universal covering. (For a detailed proof, see [9].) 52 $G$ $f$ PROPOSITION 2. Let be a locally Hamming graph with valency $r_{i}$ and let : $H(r)arrow G$ be a universal covering. Then, $d_{\pi(G,f)}\geq 5$ and $G\cong H(r)/\pi(G, f)$ hold. Thus, the classification problem of LHDRG with valency $r$ is converted to the problem to determine for which subgroup $\Gamma$ of $Aut(H(r)),$ $H(r)/\Gamma$ is a distance- regular graph. REMARK 1: Proposition 1 was proved by Brouwer[2][3, p.153], without emphasis on the universality. Almost equivalent propositions to Propositions 1 and 2 will also be found in the book by Brouwer et. a1.[3]. What is new in this paper is that we regard a covering $f$ not only as a partition but as a coset by a group $\Gamma$ . NOTE: In this paper, the letter $d$ always denotes not the diameter but the discrete- ness. \S 3. Designs. Recall that a $t-(v, k, \lambda)$ design is a family $B$ of k-element subset of a v-element set $X$ such that for any k-element subset $K$ of $X$ , the number of $B\in B$ containing $K$ is A. In Lemmas 3 and 3*, we shall prove a LHDRG produces a design with an additional property. To begin with, we classify LHDRG using the discreteness of its fundamental group. Let $LHDRG_{d}^{r}$ denote the set of LHDRGs with valency $r$ such that the discreteness of its fundamental group is equal to $d$ . Later it will be proved that a distance- regular graph belongs to $LHDRG_{d}^{r}$ for some $d\geq 7$ if and only if its parameters satisfy $a_{1}=0,$ $a_{2}=0,$ $c_{2}=2$ , and $c_{3}=3$ . Thus, the classification of LHDRG $rd$ implies the classification of distance-regular graphs with such parameters. The reason why the cases $d=5,6$ are also considered is that some interesting LHDRG with $d=5,6$ obtained from Golay codes exist, as shown in Section 5. The next lemma is useful to shorten some proofs. LEMMA 1. Let $H(r)$ be an r-dimensional Hamming scheme and let $\Gamma$ be $a$ subgroup of $d_{\Gamma}\geq 5$ $\Gamma x_{1},$ $\Gamma x_{2},$ $\Gamma x_{t}$ $H/\Gamma;i.e.,$ $\Gamma x_{j}$ $Aut(H(r))$ with . Let $\ldots$ , be a walk in is adjacent to $\Gamma x_{j+1}$ for $j=1,2,$ $\ldots t-1$ . Then, there exist $x_{2}'$ , . , $x_{t}'suclo$ that $\Gamma x_{j}'=\Gamma x_{j}$ for $j=2,$ $\ldots t$ and that $x_{1},x_{2}'$ , . , $x_{t}'$ is a walk in H. Consequently, $d_{H(r)/\Gamma}(\Gamma x, \Gamma y)=d_{H(r)}(x, \Gamma y)$ holds, where $RHS$ denotes the minimum of $d_{H(r)}(x, \gamma y)$ for all $\gamma\in\Gamma$ . $\Gamma x_{1}$ $\Gamma x_{2}$ $x_{2}'\in\Gamma x_{2}$ PROOF: Since is adjacent to , there exists an adjacent to $x_{1}$ .
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